Position of a point

Q. Acceleration of the particle in a rectilinear motion is given as $a=4t+3$ in $\left(m/s^2\right)$. The final position of the particle as a function of time $t$ is
(Given, at $t=0,u=3m/s,x=2m$)

1. $\frac{2t^3}{3}+\frac{3t^2}{2}+3t$
2. $\frac{2t^3}{3}+\frac{3t^2}{2}$
3. $\frac{2t^3}{3}+\frac{3t^2}{2}+3t+2$
4. $\frac{2t^3}{3}+\frac{3t^2}{2}-3t$

Q. A body of mass $2\text{kg}$ moves under a force of $(2\hat{i}+3\hat{j}+5\hat{k})\text{N}$. It starts from rest and was at the origin initially. After $4\text{s}$, its new coordinates are $(8,b,20)$. The value of $b$ is .
(Round off to the Nearest Integer)

Q. If position of $B$ with respect to $P$, is $+2\text{m}$ and position of $C$ with respect to $P$ is $-4\text{m}$. Where $B,P,C$ are the points along $X-$axis, then what will be the position of $B$ with respect to $C$ ?

1. $3\text{m}$
2. $6\text{m}$
3. $-5\text{m}$
4. $-2\text{m}$

Q. A body of mass $2\text{kg}$ moves under a force of $(2\hat{i}+3\hat{j}+5\hat{k})\text{N}$. It starts from rest and was at the origin initially. After $4\text{s}$, its new coordinates are $(8,b,20)$. The value of $b$ is .
(Round off to the Nearest Integer)

Q. In the given figure find
(i) position of $A$ with respect to $B$
(ii) position of $O$ with respect to $B$

1. $\left(i\right)5\left(ii\right)3$
2. $\left(i\right)-5\left(ii\right)3$
3. $\left(i\right)5\left(ii\right)-3$
4. $\left(i\right)-5\left(ii\right)-3$

Q. Instantaneous velocity of an object varies with time as $v=\alpha -\beta t^2$. Find its position, $x$ as a function of time, $t$. Also find the object`s maximum positive displacement, $x_{max}$ from the origin.

1. $x=\alpha t-\frac{\beta t^3}{3}$ and $x_{max}=\frac{2}{3}\frac{{\alpha }^{3/2}}{{\beta }^{3/2}}$
2. $x=\alpha t-\frac{\beta t^3}{3}$ and $x_{max}=\frac{2\alpha }{{\beta }^{1/2}}$
3. $x=2\alpha t-\beta$ and $x_{max}=2\alpha \beta$
4. $x=2{\alpha }^{2}t-\beta t^3$ and $x_{max}=\frac{2{\alpha }^{1/2}}{{\beta }^{3/2}}$

Q. Three balls initially at rest are released from the origin in the same direction and covers a distance of $5\text{m}$, $9\text{m}$ and $15\text{m}$ respectively. What will be the position of the ball farthest from the origin w.r.t the ball nearest to the origin?

1. $4\text{m}$
2. $6\text{m}$
3. $9\text{m}$
4. $10\text{m}$

Q. At $t=0$, a particle starts from $(1,0)$ and moves towards positive $x-$axis with speed of $v=3t^2+2t\text{m/s}$. The final position of the particle as a function of time is

1. $\frac{3t^3}{2}-2t^2+1\text{m}$
2. $\frac{3t^3}{2}+2t^2+1\text{m}$
3. $t^3+t^2+1\text{m}$
4. $t^3-t^2+1\text{m}$

Q. At $t=0$, a particle starts from $(-2,0)$ and moves towards positive $x-$axis with speed of $v=6t^2+4t\text{m/s}$. The final position of the particle and the distance travelled by the particle respectively, as a function of time are

1. $2t^3-2t^2+2\text{m},2t^3-2t^2\text{m}$
2. $2t^3+2t^2-2\text{m},2t^3+2t^2\text{m}$
3. $2t^3-2t^2-2\text{m},2t^3+2t^2\text{m}$
4. $2t^3-2t^2+2\text{m},2t^3+2t^2\text{m}$